Problem No. 1. At a local crisis center, three volunteers answer the phones to h
ID: 352704 • Letter: P
Question
Problem No. 1. At a local crisis center, three volunteers answer the phones to help solve residents' problems. Calls arrive randomly, at an average rate of 20 per eight hour day. The amount of time that a volunteer spends with a caller is exponentially distributed with and average of 40 minutes. If all the volunteers are busy, the callers are put on hold and they are assigned in the order of their arrival. a. Determine the fraction of time each volunteer is busy. b. On the average, how long is a caller put on hold? c. On the average, how many callers are on hold? d. How much time, on the average, a caller spends on phone? e. What is the mean number of callers tied up to the phone ? f. Determine the probability that all volunteers are idle. g. Find the probability that all the volunteers are busy yet no caller is on hold.Explanation / Answer
mean arrival rate,
20
per 8 hr day
2.5
per hour
service time
40
min/caller
mean service rate,
1.5
per hour
no of servers, m
3
effective service rate= m*
4.5
per hour
a. fraction of time volunteer is busy= system utilization
system utilization, =/(m)
0.56
hence the fraction of time each volunteer is busy is 56%
b. average waiting time = Wq = / (*(-))
Here, m will be used for service rate,
Wq=2.5/(4.5*(4.5-2.5))
Wq
0.28
hour
=
16.67
min
Hence, on average, a caller is put on hold for 16.67 minutes.
c. Average no in the queue, Lq=*Wq
Lq=2.5*0.28
Lq
0.69
callers
1
caller (rounding off)
Hence, on average, 1 caller (rounded off) is on hold.
d. average time in the system, W=1/(-)
W=1/(4.5-2.5)
W
0.5
hours
30
minutes
Hence, on an average, the caller spends 30 minutes on the phone.
mean arrival rate,
20
per 8 hr day
2.5
per hour
service time
40
min/caller
mean service rate,
1.5
per hour
no of servers, m
3
effective service rate= m*
4.5
per hour
a. fraction of time volunteer is busy= system utilization
system utilization, =/(m)
0.56
hence the fraction of time each volunteer is busy is 56%
b. average waiting time = Wq = / (*(-))
Here, m will be used for service rate,
Wq=2.5/(4.5*(4.5-2.5))
Wq
0.28
hour
=
16.67
min
Hence, on average, a caller is put on hold for 16.67 minutes.
c. Average no in the queue, Lq=*Wq
Lq=2.5*0.28
Lq
0.69
callers
1
caller (rounding off)
Hence, on average, 1 caller (rounded off) is on hold.
d. average time in the system, W=1/(-)
W=1/(4.5-2.5)
W
0.5
hours
30
minutes
Hence, on an average, the caller spends 30 minutes on the phone.
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